import Mathlib
import OAI.Computability.VertexCover.Information.Finite

namespace OAI

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namespace UniqueGames.Foundations.Information

open scoped BigOperators

variable {α β : Type*} [Fintype α] [Fintype β]

noncomputable def firstMarginal (p : α × β → ℝ) (a : α) : ℝ := ∑ b, p (a, b)

noncomputable def secondMarginal (p : α × β → ℝ) (b : β) : ℝ := ∑ a, p (a, b)

def product (p : α → ℝ) (q : β → ℝ) : α × β → ℝ := fun ab => p ab.1 * q ab.2

theorem firstMarginal_isProbability (p : α × β → ℝ) (hp : IsProbability p) :
    IsProbability (firstMarginal p) := by
  constructor
  · intro a
    exact Finset.sum_nonneg (fun b _ => hp.1 (a, b))
  · simpa only [firstMarginal, Fintype.sum_prod_type] using hp.2

theorem secondMarginal_isProbability (p : α × β → ℝ) (hp : IsProbability p) :
    IsProbability (secondMarginal p) := by
  constructor
  · intro b
    exact Finset.sum_nonneg (fun a _ => hp.1 (a, b))
  · change (∑ b, ∑ a, p (a, b)) = 1
    rw [Finset.sum_comm]
    exact (firstMarginal_isProbability p hp).2

theorem product_isProbability (p : α → ℝ) (q : β → ℝ)
    (hp : IsProbability p) (hq : IsProbability q) : IsProbability (product p q) := by
  constructor
  · intro ab
    exact mul_nonneg (hp.1 ab.1) (hq.1 ab.2)
  · simp only [product, Fintype.sum_prod_type, ← Finset.mul_sum, hq.2, mul_one, hp.2]

theorem point_le_firstMarginal (p : α × β → ℝ) (hp : IsProbability p) (a : α) (b : β) :
    p (a, b) ≤ firstMarginal p a := by
  classical
  exact Finset.single_le_sum (fun b _ => hp.1 (a, b)) (Finset.mem_univ b)

theorem point_le_secondMarginal (p : α × β → ℝ) (hp : IsProbability p) (a : α) (b : β) :
    p (a, b) ≤ secondMarginal p b := by
  classical
  exact Finset.single_le_sum (fun a _ => hp.1 (a, b)) (Finset.mem_univ a)

theorem joint_supportedBy_product_marginals (p : α × β → ℝ) (hp : IsProbability p) :
    SupportedBy p (product (firstMarginal p) (secondMarginal p)) := by
  intro ab hpne
  have hpos : 0 < p ab := lt_of_le_of_ne (hp.1 ab) (Ne.symm hpne)
  exact mul_ne_zero
    (ne_of_gt (hpos.trans_le (point_le_firstMarginal p hp ab.1 ab.2)))
    (ne_of_gt (hpos.trans_le (point_le_secondMarginal p hp ab.1 ab.2)))

theorem relativeEntropy_product_reference (p : α × β → ℝ) (u : α → ℝ) (v : β → ℝ)
    (hp : IsProbability p) (_hu : IsProbability u) (_hv : IsProbability v)
    (hsu : SupportedBy (firstMarginal p) u) (hsv : SupportedBy (secondMarginal p) v) :
    relativeEntropy p (product u v) =
      relativeEntropy p (product (firstMarginal p) (secondMarginal p)) +
        relativeEntropy (firstMarginal p) u + relativeEntropy (secondMarginal p) v := by
  have hpoint : ∀ ab : α × β, p ab * Real.log (p ab / product u v ab) =
      p ab * Real.log (p ab / product (firstMarginal p) (secondMarginal p) ab) +
      p ab * Real.log (firstMarginal p ab.1 / u ab.1) +
      p ab * Real.log (secondMarginal p ab.2 / v ab.2) := by
    intro ab
    by_cases hzero : p ab = 0
    · simp [hzero]
    have hpos : 0 < p ab := lt_of_le_of_ne (hp.1 ab) (Ne.symm hzero)
    have hleft := ne_of_gt (hpos.trans_le (point_le_firstMarginal p hp ab.1 ab.2))
    have hright := ne_of_gt (hpos.trans_le (point_le_secondMarginal p hp ab.1 ab.2))
    have hune := hsu ab.1 hleft
    have hvne := hsv ab.2 hright
    simp only [product, Real.log_div hzero (mul_ne_zero hune hvne),
      Real.log_div hzero (mul_ne_zero hleft hright), Real.log_mul hune hvne,
      Real.log_mul hleft hright, Real.log_div hleft hune, Real.log_div hright hvne]
    ring
  have hleftSum : (∑ ab : α × β, p ab * Real.log (firstMarginal p ab.1 / u ab.1)) =
      relativeEntropy (firstMarginal p) u := by
    simp only [Fintype.sum_prod_type, ← Finset.sum_mul, relativeEntropy, firstMarginal]
  have hrightSum : (∑ ab : α × β, p ab * Real.log (secondMarginal p ab.2 / v ab.2)) =
      relativeEntropy (secondMarginal p) v := by
    rw [Fintype.sum_prod_type, Finset.sum_comm]
    simp only [← Finset.sum_mul, relativeEntropy, secondMarginal]
  unfold relativeEntropy
  simp_rw [hpoint]
  rw [Finset.sum_add_distrib, Finset.sum_add_distrib, hleftSum, hrightSum]
  rfl

theorem marginal_relativeEntropy_sum_le (p : α × β → ℝ) (u : α → ℝ) (v : β → ℝ)
    (hp : IsProbability p) (hu : IsProbability u) (hv : IsProbability v)
    (hsu : SupportedBy (firstMarginal p) u) (hsv : SupportedBy (secondMarginal p) v) :
    relativeEntropy (firstMarginal p) u + relativeEntropy (secondMarginal p) v ≤
      relativeEntropy p (product u v) := by
  rw [relativeEntropy_product_reference p u v hp hu hv hsu hsv]
  have hn := relativeEntropy_nonneg p (product (firstMarginal p) (secondMarginal p)) hp
    (product_isProbability _ _ (firstMarginal_isProbability p hp)
      (secondMarginal_isProbability p hp)) (joint_supportedBy_product_marginals p hp)
  linarith

end UniqueGames.Foundations.Information

end


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end OAI